Quenched Survival of Bernoulli Percolation on Galton–Watson Trees

Abstract

We explore the survival function for percolation on Galton–Watson trees. Letting g(T, p) represent the probability a tree T survives Bernoulli percolation with parameter p, we establish several results about the behavior of the random function $$g(\mathbf{T}, \cdot )$$ , where $$\mathbf{T}$$ is drawn from the Galton–Watson distribution. These include almost sure smoothness in the supercritical region; an expression for the $$k\text {th}$$ -order Taylor expansion of $$g(\mathbf{T}, \cdot )$$ at criticality in terms of limits of martingales defined from $$\mathbf{T}$$ (this requires a moment condition depending on k); and a proof that the $$k\text {th}$$ order derivative extends continuously to the critical value. Each of these results is shown to hold for almost every Galton–Watson tree.